Question

If $A_1,A_2,G_1,G_2$ and $H_1,H_2$ are two A.Ms, G.Ms and H.M.s between two numbers, then $\frac{A_1+A_2}{H_1+H_2}.\frac{H_1{H}_2}{G_1{G}_2}$ equals to.

• A. 0
• B. 4
• C. 1
• D. 8

Answer 1

The correct option is C 1

$A_1,A_2,G_1,G_2$ and $H_1,H_2$ are two A.Ms, G.Ms and H.M.s between $a$ and $b$.
$a,A_1,A_2,b$ are in AP $\Rightarrow A_1+A_2=a+b$
$a,G_1,G_2,b$ are in GP $\Rightarrow G_1{G}_2=ab$
$a,H_1,H_2,b$ are in HP $\Rightarrow \frac{1}{H_1}+\frac{1}{H_2}=\frac{1}{a}+\frac{1}{b}$
$\Rightarrow \frac{H_1{H}_2}{H_1+H_2}=\frac{ab}{a+b}$
$\frac{A_1+A_2}{H_1+H_2}.\frac{H_1{H}_2}{G_1{G}_2}$$=\frac{a+b}{a+b}\cdot \frac{ab}{ab}=1$
Hence, option C.